How many wires are in the box where the switch that controls the lamp in the utility room? You have to find a point where there is always power, not switched power. Then the rest is fairly straightforward...need to know what you have first, though.

You need to run a circuit (hot + neutral + ground) from some point to either (1) the new switch, or (2) the new light. The existing light apparently has that circuit available, but you can get it from somewhere else if that is more convenient.

(1) If power is available at the new switch:
a. Run a cable (hot + neutral + ground) from the switch to the new light.
b. In the switch box; (i) connect the white from the power cable to the white running to the new light (ii) Connect the green to the box or switch ground and to the green that is running to the new light, (iii) connect the black from the power cable to one of the switch terminals, and (iv) connect the black in the cable to the light to the other terminal of the switch.
c. At the light; connect white to white, black to black, and green to ground.

(2) If power is available at the new light:
a. Run a cable (hot + neutral + ground) from the switch to the new light.
b. At the light: (i) Connect the power source cable white to the white of the light (ii) Connect the green of the power cable to the ground of the light, the box if it is metal, and the green of the cable to the switch, (iii) Connect the black conductor of the supply cable to the white conductor going to the switch, and (iv) Connect the black wire returning from the switch to the black conductor of the light.
c. At the switch: (i) Connect the white wire from the light to one terminal of the switch. (ii) Connect the black conductor from the light to the other terminal of the switch, and (iii) Connect the green wire to any metal box and/or the ground of the switch.